3.16 \(\int \frac{(c+d x)^3}{a+i a \cot (e+f x)} \, dx\)

Optimal. Leaf size=189 \[ \frac{3 i d^2 (c+d x)}{4 f^3 (a+i a \cot (e+f x))}+\frac{3 d (c+d x)^2}{4 f^2 (a+i a \cot (e+f x))}-\frac{i (c+d x)^3}{2 f (a+i a \cot (e+f x))}-\frac{3 d (c+d x)^2}{8 a f^2}+\frac{i (c+d x)^3}{4 a f}+\frac{(c+d x)^4}{8 a d}-\frac{3 d^3}{8 f^4 (a+i a \cot (e+f x))}-\frac{3 i d^3 x}{8 a f^3} \]

[Out]

(((-3*I)/8)*d^3*x)/(a*f^3) - (3*d*(c + d*x)^2)/(8*a*f^2) + ((I/4)*(c + d*x)^3)/(a*f) + (c + d*x)^4/(8*a*d) - (
3*d^3)/(8*f^4*(a + I*a*Cot[e + f*x])) + (((3*I)/4)*d^2*(c + d*x))/(f^3*(a + I*a*Cot[e + f*x])) + (3*d*(c + d*x
)^2)/(4*f^2*(a + I*a*Cot[e + f*x])) - ((I/2)*(c + d*x)^3)/(f*(a + I*a*Cot[e + f*x]))

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Rubi [A]  time = 0.200088, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3723, 3479, 8} \[ \frac{3 i d^2 (c+d x)}{4 f^3 (a+i a \cot (e+f x))}+\frac{3 d (c+d x)^2}{4 f^2 (a+i a \cot (e+f x))}-\frac{i (c+d x)^3}{2 f (a+i a \cot (e+f x))}-\frac{3 d (c+d x)^2}{8 a f^2}+\frac{i (c+d x)^3}{4 a f}+\frac{(c+d x)^4}{8 a d}-\frac{3 d^3}{8 f^4 (a+i a \cot (e+f x))}-\frac{3 i d^3 x}{8 a f^3} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(a + I*a*Cot[e + f*x]),x]

[Out]

(((-3*I)/8)*d^3*x)/(a*f^3) - (3*d*(c + d*x)^2)/(8*a*f^2) + ((I/4)*(c + d*x)^3)/(a*f) + (c + d*x)^4/(8*a*d) - (
3*d^3)/(8*f^4*(a + I*a*Cot[e + f*x])) + (((3*I)/4)*d^2*(c + d*x))/(f^3*(a + I*a*Cot[e + f*x])) + (3*d*(c + d*x
)^2)/(4*f^2*(a + I*a*Cot[e + f*x])) - ((I/2)*(c + d*x)^3)/(f*(a + I*a*Cot[e + f*x]))

Rule 3723

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + d*x)^(m + 1)/(2*
a*d*(m + 1)), x] + (Dist[(a*d*m)/(2*b*f), Int[(c + d*x)^(m - 1)/(a + b*Tan[e + f*x]), x], x] - Simp[(a*(c + d*
x)^m)/(2*b*f*(a + b*Tan[e + f*x])), x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0] && GtQ[m, 0]

Rule 3479

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*(a + b*Tan[c + d*x])^n)/(2*b*d*n), x] +
Dist[1/(2*a), Int[(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && LtQ[n
, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{(c+d x)^3}{a+i a \cot (e+f x)} \, dx &=\frac{(c+d x)^4}{8 a d}-\frac{i (c+d x)^3}{2 f (a+i a \cot (e+f x))}+\frac{(3 i d) \int \frac{(c+d x)^2}{a+i a \cot (e+f x)} \, dx}{2 f}\\ &=\frac{i (c+d x)^3}{4 a f}+\frac{(c+d x)^4}{8 a d}+\frac{3 d (c+d x)^2}{4 f^2 (a+i a \cot (e+f x))}-\frac{i (c+d x)^3}{2 f (a+i a \cot (e+f x))}-\frac{\left (3 d^2\right ) \int \frac{c+d x}{a+i a \cot (e+f x)} \, dx}{2 f^2}\\ &=-\frac{3 d (c+d x)^2}{8 a f^2}+\frac{i (c+d x)^3}{4 a f}+\frac{(c+d x)^4}{8 a d}+\frac{3 i d^2 (c+d x)}{4 f^3 (a+i a \cot (e+f x))}+\frac{3 d (c+d x)^2}{4 f^2 (a+i a \cot (e+f x))}-\frac{i (c+d x)^3}{2 f (a+i a \cot (e+f x))}-\frac{\left (3 i d^3\right ) \int \frac{1}{a+i a \cot (e+f x)} \, dx}{4 f^3}\\ &=-\frac{3 d (c+d x)^2}{8 a f^2}+\frac{i (c+d x)^3}{4 a f}+\frac{(c+d x)^4}{8 a d}-\frac{3 d^3}{8 f^4 (a+i a \cot (e+f x))}+\frac{3 i d^2 (c+d x)}{4 f^3 (a+i a \cot (e+f x))}+\frac{3 d (c+d x)^2}{4 f^2 (a+i a \cot (e+f x))}-\frac{i (c+d x)^3}{2 f (a+i a \cot (e+f x))}-\frac{\left (3 i d^3\right ) \int 1 \, dx}{8 a f^3}\\ &=-\frac{3 i d^3 x}{8 a f^3}-\frac{3 d (c+d x)^2}{8 a f^2}+\frac{i (c+d x)^3}{4 a f}+\frac{(c+d x)^4}{8 a d}-\frac{3 d^3}{8 f^4 (a+i a \cot (e+f x))}+\frac{3 i d^2 (c+d x)}{4 f^3 (a+i a \cot (e+f x))}+\frac{3 d (c+d x)^2}{4 f^2 (a+i a \cot (e+f x))}-\frac{i (c+d x)^3}{2 f (a+i a \cot (e+f x))}\\ \end{align*}

Mathematica [A]  time = 0.596639, size = 246, normalized size = 1.3 \[ \frac{i (\cos (2 e)+i \sin (2 e)) \cos (2 f x) \left (6 c^2 d f^2 (2 f x+i)+4 c^3 f^3+6 c d^2 f \left (2 f^2 x^2+2 i f x-1\right )+d^3 \left (4 f^3 x^3+6 i f^2 x^2-6 f x-3 i\right )\right )-(\cos (2 e)+i \sin (2 e)) \sin (2 f x) \left (6 c^2 d f^2 (2 f x+i)+4 c^3 f^3+6 c d^2 f \left (2 f^2 x^2+2 i f x-1\right )+d^3 \left (4 f^3 x^3+6 i f^2 x^2-6 f x-3 i\right )\right )+2 f^4 x \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right )}{16 a f^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3/(a + I*a*Cot[e + f*x]),x]

[Out]

(2*f^4*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) + I*(4*c^3*f^3 + 6*c^2*d*f^2*(I + 2*f*x) + 6*c*d^2*f*(-1
+ (2*I)*f*x + 2*f^2*x^2) + d^3*(-3*I - 6*f*x + (6*I)*f^2*x^2 + 4*f^3*x^3))*Cos[2*f*x]*(Cos[2*e] + I*Sin[2*e])
- (4*c^3*f^3 + 6*c^2*d*f^2*(I + 2*f*x) + 6*c*d^2*f*(-1 + (2*I)*f*x + 2*f^2*x^2) + d^3*(-3*I - 6*f*x + (6*I)*f^
2*x^2 + 4*f^3*x^3))*(Cos[2*e] + I*Sin[2*e])*Sin[2*f*x])/(16*a*f^4)

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Maple [A]  time = 0.199, size = 170, normalized size = 0.9 \begin{align*}{\frac{{d}^{3}{x}^{4}}{8\,a}}+{\frac{c{d}^{2}{x}^{3}}{2\,a}}+{\frac{3\,{c}^{2}d{x}^{2}}{4\,a}}+{\frac{{c}^{3}x}{2\,a}}+{\frac{{c}^{4}}{8\,ad}}+{\frac{{\frac{i}{16}} \left ( 4\,{d}^{3}{x}^{3}{f}^{3}+6\,i{d}^{3}{f}^{2}{x}^{2}+12\,c{d}^{2}{f}^{3}{x}^{2}+12\,ic{d}^{2}{f}^{2}x+12\,{c}^{2}d{f}^{3}x+6\,i{c}^{2}d{f}^{2}+4\,{c}^{3}{f}^{3}-6\,{d}^{3}fx-3\,i{d}^{3}-6\,c{d}^{2}f \right ){{\rm e}^{2\,i \left ( fx+e \right ) }}}{a{f}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3/(a+I*a*cot(f*x+e)),x)

[Out]

1/8/a*d^3*x^4+1/2/a*d^2*c*x^3+3/4/a*d*c^2*x^2+1/2/a*c^3*x+1/8/a/d*c^4+1/16*I*(4*d^3*x^3*f^3+6*I*d^3*f^2*x^2+12
*c*d^2*f^3*x^2+12*I*c*d^2*f^2*x+12*c^2*d*f^3*x+6*I*c^2*d*f^2+4*c^3*f^3-6*d^3*f*x-3*I*d^3-6*c*d^2*f)/a/f^4*exp(
2*I*(f*x+e))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+I*a*cot(f*x+e)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.61158, size = 343, normalized size = 1.81 \begin{align*} \frac{2 \, d^{3} f^{4} x^{4} + 8 \, c d^{2} f^{4} x^{3} + 12 \, c^{2} d f^{4} x^{2} + 8 \, c^{3} f^{4} x +{\left (4 i \, d^{3} f^{3} x^{3} + 4 i \, c^{3} f^{3} - 6 \, c^{2} d f^{2} - 6 i \, c d^{2} f + 3 \, d^{3} +{\left (12 i \, c d^{2} f^{3} - 6 \, d^{3} f^{2}\right )} x^{2} +{\left (12 i \, c^{2} d f^{3} - 12 \, c d^{2} f^{2} - 6 i \, d^{3} f\right )} x\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{16 \, a f^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+I*a*cot(f*x+e)),x, algorithm="fricas")

[Out]

1/16*(2*d^3*f^4*x^4 + 8*c*d^2*f^4*x^3 + 12*c^2*d*f^4*x^2 + 8*c^3*f^4*x + (4*I*d^3*f^3*x^3 + 4*I*c^3*f^3 - 6*c^
2*d*f^2 - 6*I*c*d^2*f + 3*d^3 + (12*I*c*d^2*f^3 - 6*d^3*f^2)*x^2 + (12*I*c^2*d*f^3 - 12*c*d^2*f^2 - 6*I*d^3*f)
*x)*e^(2*I*f*x + 2*I*e))/(a*f^4)

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Sympy [A]  time = 0.772265, size = 360, normalized size = 1.9 \begin{align*} \begin{cases} \frac{\left (4 i a^{3} c^{3} f^{9} e^{2 i e} + 12 i a^{3} c^{2} d f^{9} x e^{2 i e} - 6 a^{3} c^{2} d f^{8} e^{2 i e} + 12 i a^{3} c d^{2} f^{9} x^{2} e^{2 i e} - 12 a^{3} c d^{2} f^{8} x e^{2 i e} - 6 i a^{3} c d^{2} f^{7} e^{2 i e} + 4 i a^{3} d^{3} f^{9} x^{3} e^{2 i e} - 6 a^{3} d^{3} f^{8} x^{2} e^{2 i e} - 6 i a^{3} d^{3} f^{7} x e^{2 i e} + 3 a^{3} d^{3} f^{6} e^{2 i e}\right ) e^{2 i f x}}{16 a^{4} f^{10}} & \text{for}\: 16 a^{4} f^{10} \neq 0 \\- \frac{c^{3} x e^{2 i e}}{2 a} - \frac{3 c^{2} d x^{2} e^{2 i e}}{4 a} - \frac{c d^{2} x^{3} e^{2 i e}}{2 a} - \frac{d^{3} x^{4} e^{2 i e}}{8 a} & \text{otherwise} \end{cases} + \frac{c^{3} x}{2 a} + \frac{3 c^{2} d x^{2}}{4 a} + \frac{c d^{2} x^{3}}{2 a} + \frac{d^{3} x^{4}}{8 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3/(a+I*a*cot(f*x+e)),x)

[Out]

Piecewise(((4*I*a**3*c**3*f**9*exp(2*I*e) + 12*I*a**3*c**2*d*f**9*x*exp(2*I*e) - 6*a**3*c**2*d*f**8*exp(2*I*e)
+ 12*I*a**3*c*d**2*f**9*x**2*exp(2*I*e) - 12*a**3*c*d**2*f**8*x*exp(2*I*e) - 6*I*a**3*c*d**2*f**7*exp(2*I*e)
+ 4*I*a**3*d**3*f**9*x**3*exp(2*I*e) - 6*a**3*d**3*f**8*x**2*exp(2*I*e) - 6*I*a**3*d**3*f**7*x*exp(2*I*e) + 3*
a**3*d**3*f**6*exp(2*I*e))*exp(2*I*f*x)/(16*a**4*f**10), Ne(16*a**4*f**10, 0)), (-c**3*x*exp(2*I*e)/(2*a) - 3*
c**2*d*x**2*exp(2*I*e)/(4*a) - c*d**2*x**3*exp(2*I*e)/(2*a) - d**3*x**4*exp(2*I*e)/(8*a), True)) + c**3*x/(2*a
) + 3*c**2*d*x**2/(4*a) + c*d**2*x**3/(2*a) + d**3*x**4/(8*a)

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Giac [A]  time = 1.21764, size = 328, normalized size = 1.74 \begin{align*} \frac{2 \, d^{3} f^{4} x^{4} + 8 \, c d^{2} f^{4} x^{3} + 4 i \, d^{3} f^{3} x^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 12 \, c^{2} d f^{4} x^{2} + 12 i \, c d^{2} f^{3} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 8 \, c^{3} f^{4} x + 12 i \, c^{2} d f^{3} x e^{\left (2 i \, f x + 2 i \, e\right )} - 6 \, d^{3} f^{2} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 4 i \, c^{3} f^{3} e^{\left (2 i \, f x + 2 i \, e\right )} - 12 \, c d^{2} f^{2} x e^{\left (2 i \, f x + 2 i \, e\right )} - 6 \, c^{2} d f^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 6 i \, d^{3} f x e^{\left (2 i \, f x + 2 i \, e\right )} - 6 i \, c d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + 3 \, d^{3} e^{\left (2 i \, f x + 2 i \, e\right )}}{16 \, a f^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+I*a*cot(f*x+e)),x, algorithm="giac")

[Out]

1/16*(2*d^3*f^4*x^4 + 8*c*d^2*f^4*x^3 + 4*I*d^3*f^3*x^3*e^(2*I*f*x + 2*I*e) + 12*c^2*d*f^4*x^2 + 12*I*c*d^2*f^
3*x^2*e^(2*I*f*x + 2*I*e) + 8*c^3*f^4*x + 12*I*c^2*d*f^3*x*e^(2*I*f*x + 2*I*e) - 6*d^3*f^2*x^2*e^(2*I*f*x + 2*
I*e) + 4*I*c^3*f^3*e^(2*I*f*x + 2*I*e) - 12*c*d^2*f^2*x*e^(2*I*f*x + 2*I*e) - 6*c^2*d*f^2*e^(2*I*f*x + 2*I*e)
- 6*I*d^3*f*x*e^(2*I*f*x + 2*I*e) - 6*I*c*d^2*f*e^(2*I*f*x + 2*I*e) + 3*d^3*e^(2*I*f*x + 2*I*e))/(a*f^4)